The Bubnov-Galerkin method is the most widely used weighted average method. This method is the basis of most finite element methods.
The finite-dimensional Galerkin form of the problem statement of our second order ODE is :
Since the basis functions () are known and linearly independent, the approximate solution is completely determined once the constants () are known.
The Galerkin method provides a great way of constructing solutions. But the question is: how do we choose so that these functions are not only linearly independent but arbitrary? Since the solution is expressed as a sum of these functions, the accuracy of our result depends strongly on the choice of .
Let the trial solution take the form,
According to the Bubnov-Galerkin approach, the weighting function also takes a similar form
Plug these values into the weak form to get
Taking the sums and constants outside the integrals and rearranging, we get
Since the s are arbitrary, the quantity inside the square brackets must be zero. That is
Let us define
Then we get a set of simultaneous linear equations